Problem:
At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two professors from each of the three departments. Find the number of possible committees that can be formed subject to these requirements.
Solution:
If exactly one male professor is chosen from each department, then one female professor from each department must also be chosen. There are ways to form the committee in this manner. If two male professors are chosen from one of the departments, then two female professors must be chosen from one of the other two departments. The committee must then be completed by choosing one of the two male professors and one of the two female professors from the remaining department. There are ways to form the committee in this manner. Thus the number of committees that meet the requirements is .
The problems on this page are the property of the MAA's American Mathematics Competitions